Orbital magnetoelectric coupling at finite electric field

We extend the band theory of linear orbital magnetoelectric coupling to treat crystals under finite electric fields. Previous work established that the orbital magnetoelectric response of a generic insulator at zero field comprises three contributions that were denoted as local circulation, itinerant circulation, and Chern-Simons. We find that the expression for each of them is modified by the presence of a dc electric field. Remarkably, the sum of the three correction terms vanishes, so that the total coupling is still given by the same formula as at zero field. This conclusion is confirmed by numerical tests on a tight-binding model, for which we calculate the field-induced change in the linear magnetoelectric coefficient.

Figure 1

Decomposition of ${\gamma }_{\mathit{zzz}}$ of Eq. (19) into $\gamma {}^{\mathrm{LC}}$ (solid lines), $\gamma {}^{\mathrm{IC}}$ (dashed lines), and $\gamma {}^{\mathrm{CS}}$ (dotted lines) calculated using Eqs. (20) and (21). Symbols denote the same contributions evaluated using Eq. (22).

Figure 2

(a) Right-hand side (solid line) and left-hand side (symbols) of Eq. (23). Squares and circles denote the LC and IC contributions, respectively. (b) Equation (20) (solid line) and Eq. (22) for the CS contribution (crosses), both multiplied by a factor of $-1/4$ for visual check of Eq. (18) by comparison to (a).